Linear elimination in chemical reaction networks

Meritxell Sáez, Elisenda Feliu, Carsten Wiuf

Research output: Book chapterChapterpeer-review


We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of ℝn. The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.

Original languageEnglish
Title of host publicationSEMA SIMAI Springer Series
PublisherSpringer International Publishing
Number of pages17
Publication statusPublished - 2019
Externally publishedYes

Publication series

NameSEMA SIMAI Springer Series
ISSN (Print)2199-3041
ISSN (Electronic)2199-305X


  • Conservation law
  • Elimination
  • Linear system
  • Noninteracting
  • Reaction network
  • Steady states


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